In previous works we derived equations for the average momentum of high-energy electrons experiencing quantum radiation reaction in strong electromagnetic plane-wave background fields. In this paper we derive similar equations for the momentum spectrum. We formulate the equations in terms of the cumulative function and study the relation between the equations for the spectrum and the equations for the moments, analyze the structure of the low-energy expansions, and finally explain how our formulation is essentially in terms of a "Green's function"which allows us to study the dynamics without choosing a specific initial wave packet (or particle-bunch distribution).